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### A convenient setting for differential geometry and global analysis II

Cahiers de Topologie et Géométrie Différentielle Catégoriques

### A general topology approach to the study of differentiability of convex functions in Banach spaces

Abstracta. 6th Winter School on Abstract Analysis

### A global differentiability result for solutions of nonlinear elliptic problems with controlled growths

Czechoslovak Mathematical Journal

Let $\Omega$ be a bounded open subset of ${ℝ}^{n}$, $n>2$. In $\Omega$ we deduce the global differentiability result $u\in {H}^{2}\left(\Omega ,{ℝ}^{N}\right)$ for the solutions $u\in {H}^{1}\left(\Omega ,{ℝ}^{n}\right)$ of the Dirichlet problem $u-g\in {H}_{0}^{1}\left(\Omega ,{ℝ}^{N}\right),-\sum _{i}{D}_{i}{a}^{i}\left(x,u,Du\right)={B}_{0}\left(x,u,Du\right)$ with controlled growth and nonlinearity $q=2$. The result was obtained by first extending the interior differentiability result near the boundary and then proving the global differentiability result making use of a covering procedure.

### A Hahn-Banach extension theorem for analytic mappings

Bulletin de la Société Mathématique de France

### A nonsmooth exponential

Studia Mathematica

Let ℳ be a type II₁ von Neumann algebra, τ a trace in ℳ, and L²(ℳ,τ) the GNS Hilbert space of τ. If L²(ℳ,τ)₊ is the completion of the set ${ℳ}_{sa}$ of selfadjoint elements, then each element ξ ∈ L²(ℳ,τ)₊ gives rise to a selfadjoint unbounded operator ${L}_{\xi }$ on L²(ℳ,τ). In this note we show that the exponential exp: L²(ℳ,τ)₊ → L²(ℳ,τ), $exp\left(\xi \right)={e}^{i{L}_{\xi }}$, is continuous but not differentiable. The same holds for the Cayley transform $C\left(\xi \right)=\left({L}_{\xi }-i\right){\left({L}_{\xi }+i\right)}^{-1}$. We also show that the unitary group ${U}_{ℳ}\subset L²\left(ℳ,\tau \right)$ with the strong operator topology is not an embedded submanifold...

### Abgeschlossene Garben differenzierbarer Funktionen.

Manuscripta mathematica

### Analysis over infinite dimensional spaces and applications to quantum field theory

Séminaire Jean Leray

### Banachian Differentiable Spaces.

Mathematische Annalen

### Banach-Stone Theorems for Banach Manifolds.

Revista de la Real Academia de Ciencias Exactas Físicas y Naturales

### Bayoumi Quasi-Differential is different from Fréchet-Differential

Open Mathematics

We prove that the Quasi Differential of Bayoumi of maps between locally bounded F-spaces may not be Fréchet-Differential and vice versa. So a new concept has been discovered with rich applications (see [1–6]). Our F-spaces here are not necessarily locally convex

### Connections on infinite dimensional manifolds with corners

Rendiconti del Seminario Matematico della Università di Padova

### Connections on some functional bundles

Czechoslovak Mathematical Journal

### Cross-sections of solution funnels in Banach spaces

Studia Mathematica

### Differentiable manifolds with generalized boundary

Czechoslovak Mathematical Journal

### Embedding of Hilbert manifolds with smooth boundary into semispaces of Hilbert spaces

Archivum Mathematicum

In this paper we prove the existence of a closed neat embedding of a Hausdorff paracompact Hilbert manifold with smooth boundary into $H×\left[0,+\infty \right)$, where $H$ is a Hilbert space, such that the normal space in each point of a certain neighbourhood of the boundary is contained in $H×\left\{0\right\}$. Then, we give a neccesary and sufficient condition that a Hausdorff paracompact topological space could admit a differentiable structure of class $\infty$ with smooth boundary.

### Every paracompact ${C}^{m}$-manifold modelled on the infinite countable product of lines is ${C}^{m}$-stable

Colloquium Mathematicae

### Extension of smooth functions in infinite dimensions II: manifolds

Studia Mathematica

Let M be a separable ${C}^{\infty }$ Finsler manifold of infinite dimension. Then it is proved, amongst other results, that under suitable conditions of local extensibility the germ of a ${C}^{\infty }$ function, or of a ${C}^{\infty }$ section of a vector bundle, on the union of a closed submanifold and a closed locally compact set in M, extends to a ${C}^{\infty }$ function on the whole of M.

### Implicit functions from locally convex spaces to Banach spaces

Studia Mathematica

We first generalize the classical implicit function theorem of Hildebrandt and Graves to the case where we have a Keller ${C}_{\Pi }^{k}$-map f defined on an open subset of E×F and with values in F, for E an arbitrary Hausdorff locally convex space and F a Banach space. As an application, we prove that under a certain transversality condition the preimage of a submanifold is a submanifold for a map from a Fréchet manifold to a Banach manifold.

### Induced differential forms on manifolds of functions

Archivum Mathematicum

Differential forms on the Fréchet manifold $ℱ\left(S,M\right)$ of smooth functions on a compact $k$-dimensional manifold $S$ can be obtained in a natural way from pairs of differential forms on $M$ and $S$ by the hat pairing. Special cases are the transgression map ${\Omega }^{p}\left(M\right)\to {\Omega }^{p-k}\left(ℱ\left(S,M\right)\right)$ (hat pairing with a constant function) and the bar map ${\Omega }^{p}\left(M\right)\to {\Omega }^{p}\left(ℱ\left(S,M\right)\right)$ (hat pairing with a volume form). We develop a hat calculus similar to the tilda calculus for non-linear Grassmannians .

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